%I #18 Feb 05 2021 18:18:44
%S 1,2,3,4,5,7,8,9,11,13,14,16,17,19,21,23,25,26,28,30,31,33,35,37,39,
%T 41,44,46,47,49,51,53,55,56,58,60,62,64,66,69,71,73,76,79,81,84,85,87,
%U 89,91,93,95,98,100,101,103,105,107,109,111,114,116,118,121,124
%N Number of terms in the greedy Lucas-and-Fibonacci representations of 1,2,...,n; partial sums of A214973.
%C For comparison with Zeckendorf (Fibonacci) representations, it is conjectured that the limit of A179180(n)/A214981(n) exists and is between 1.2 and 1.4.
%H Clark Kimberling, <a href="/A214981/b214981.txt">Table of n, a(n) for n = 1..10000</a>
%H Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Kimberling/kimber12.html">Lucas Representations of Positive Integers</a>, J. Int. Seq., Vol. 23 (2020), Article 20.9.5.
%e The basis is B = (1,2,3,4,5,7,8,11,13,18,21,29,34,47,55,...), composed of Fibonacci numbers and Lucas numbers. Representations of positive integers using the greedy algorithm on B:
%e n repres. # terms a(n)
%e 1 1 1 1
%e 2 2 1 2
%e 3 3 1 3
%e 4 4 1 4
%e 5 5 1 5
%e 6 5+1 2 7
%e 7 7 1 8
%e 8 8 1 9
%e 9 8+1 2 11
%e 10 8+2 2 13
%e 27 21+5+1 3 44
%t (See the program at A214973.)
%Y Cf. A000032, A000045, A179180, A214973, A214977, A214979, A214980, A214981.
%K nonn
%O 1,2
%A _Clark Kimberling_, Oct 22 2012
%E Edited by _Clark Kimberling_, Jun 13 2020
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